Download 2 is

Physics 102: Lecture 25
At i Spectroscopy
Atomic
S t
& Quantum
Q t Atoms
At
Physics 102: Lecture 25, Slide 1
From last lecture – Bohr model
Angular momentum is quantized
Ln = nh/2π
n = 1, 2, 3 ...
Energy is quantized
mk 2 e 4 Z 2
13.6 ⋅ Z 2
En = −
≈−
eV ( where h ≡ h / 2π )
2
2
2
2h n
n
Radius is quantized
2
2
n2
⎛ h ⎞ 1 n
= ( 0.0529 nm )
rn = ⎜
⎟
2
Z
⎝ 2π ⎠ mke Z
Velocity too! En= -13.6 Z2/n2 = ½ mvn2
Physics 102: Lecture 25, Slide 2
+Ze
Transitions + Energy Conservation
• Each orbit has a specific energy:
energ :
En= -13.6 Z2/n2
• Photon absorbed when electron
j
jumps
from
f
low
l energy to
t high
hi h
energy. Photon emitted when
electron
l t
jumps
j
from
f
high
hi h energy to
t
low energy orbit:
E2 – E1 = h f = h c / λ
Physics 102: Lecture 25, Slide 3
E2
E1
Demo: Line Spectra
p
In addition to the continuous blackbody spectrum,
elements emit a discrete set of wavelengths which
show up as lines in a diffraction grating.
En= -13.6 Z2/n2
656 nm
H
n=3
n=2
This is how neon signs &
Na lamps work!
Spectra give us information on
atomic
t i structure
t
t
Physics 102: Lecture 25, Slide 4
n=1
Checkpoint 1.1
Electron A falls from energy level n=2 to energy level n=1
(
(ground
d state),
t t ) causing
i a photon
h t to
t be
b emitted.
itt d
Electron B falls from energy level nn=33 to energy level nn=11
(ground state), causing a photon to be emitted.
n=3
n=2
Which photon has more energy?
1) Photon A
2) Photon B
A
B
n=1
Physics 102: Lecture 25, Slide 5
Spectral Line Wavelengths
Calculate the wavelength of photon emitted when an electron in the
hydrogen
yd oge atom
ato ddrops
ops from
o tthee n=2 state to the
t e ground
g ou d state ((n=1).
).
E2= -3.4 eV
n=3
n=2
Z2
E n = −13.6eV 2
n
hf = E2 − E1
= −3.4eV − (−13.6eV) = 10.2eV
E1= -13.6 eV
Ephoton =
Physics 102: Lecture 25, Slide 6
n=1
hc
λ
hc
1240
λ=
=
≈ 124nm
10.2eV 10.2
ACT: Spectral Line Wavelengths
Compare the wavelength of a photon produced from a transition
from
o n=3
3 to n=2 with
w t that
t at of
o a photon
p oto produced
p oduced from
o a transition
ta sto
n=2 to n=1.
(Α)
λ32 < λ21
(Β)
λ32 = λ21
(C)
λ32 > λ21
E32 < E21
Physics 102: Lecture 25, Slide 7
so
n=3
3
n=2
λ32 > λ21
n=1
ACT/Checkpoint 1.2
The electrons in a large group of hydrogen atoms are
eexcited
c ted to the
t e n=33 level.
eve . How
ow many
a y spectral
spect a lines
es will
w
be produced?
A. 1
B. 2
n=3
n=2
C 3
C.
D. 4
E 5
E.
n=1
Physics 102: Lecture 25, Slide 8
The Bohr Model is incorrect!
To be consistent with the Heisenberg Uncertainty Principle
Principle, which
of these properties cannot be quantized (have the exact value
known)?
Electron Radius
Would know location
Electron Energy
Electron Velocity
Would know momentum
Electron Angular Momentum
B in
But,
i the
h Bohr
B h model:
d l
2
2
2
h
1
n
n
⎛
⎞
rn = ⎜
= ( 0.0529
0 0529 nm )
⎟
2
Z
⎝ 2π ⎠ mke Z
Physics 102: Lecture 25, Slide 9
Quantized radii
and velocities for
electron orbitals
Checkpoint 2
+Ze
Bohr Model
Quantum Atom
So what keeps the electron from “sticking” to the nucleus?
Centripetal Acceleration
Pauli Exclusion Principle
Heisenberg Uncertainty Principle
Physics 102: Lecture 25, Slide 10
Quantum Mechanics
Q
Theory used to predict probability distributions
QM
Physics 102: Lecture 25, Slide 11
Quantum Mechanical Atom
• Predicts available energy states agreeing with
Bohr.
Bohr
• Don’t have definite electron position, only a
probability
b bilit function.
f ti
• Each orbital can have 0 angular momentum!
• Each electron state labeled by 4 numbers:
n = pprincipal
p quantum
q
number (1,
( 2, 3, …))
l = angular momentum (0, 1, 2, … n-1)
p
of l ((-l < ml < l))
ml = component
ms = spin (-½ , +½)
Physics 102: Lecture 25, Slide 12
Quantum Mechanics (vs. Bohr)
Electrons are described by a probability function,
not a definite radius!
Physics 102: Lecture 25, Slide 13
Quantum Numbers
Each electron in an atom is labeled by 4 #’s
n = Principal
p Q
Quantum Number ((1,, 2,, 3,, …))
• Determines the Bohr energy
l = Orbital Quantum Number (0,
(0 1,
1 2,
2 … n-1)
n 1)
• Determines angular momentum
• l <n
always true!
h
L = l(l + 1)
2π
ml = Magnetic Quantum Number (-l , … 0, … l )
• zz-component
component of l
• | ml | <= l
always true!
ms = S
Spin
i Q
Quantum
t
N
Number
b ((-½
½ , +½)
• “Up Spin” or “Down Spin”
Physics 102: Lecture 25, Slide 14
h
Lz = ml
2π
ACT: Quantum numbers
For which state of hydrogen is the orbital
angular momentum required to be zero?
1. n=1
2. n=2
3. n=3
Physics 102: Lecture 25, Slide 15
The allowed values of l are
0, 1, 2, …, n-1. When n=1, l
must be zero.
Spectroscopic
p
p Nomenclature
“Shells”
“Subshells”
l =0
0 is “s
s state
state”
l =1 is “p state”
l =2 is “d state”
l =3 is “f state”
l =4 is “g
g state”
n=1 is “K
K shell
shell”
n=2 is “L shell”
n=3
n
3 is “M
M shell
shell”
n=4 is “N shell”
n=5
n
5 is “O
O shell
shell”
1 electron in ground state of Hydrogen:
n=1, l =0 is denoted as: 1s1
n=1
Physics 102: Lecture 25, Slide 16
l =0
1 electron
Electron orbitals
IIn correct quantum mechanical
h i l description
d
i i off atoms, positions
ii
off
electrons not quantized, orbitals represent probabilities
Carbon orbitals
imaged in 2009 using
electron microscopy!
Physics 102: Lecture 25, Slide 17
Quantum Numbers
How many unique electron states exist with n
n=2?
2?
l = 0 : 2s2
ml = 0 : ms = ½ , -½
2 states
l = 1 : 2p
p6
ml = +1: ms = ½ , -½
ml = 0: ms = ½ , -½
ml = -1:
1 ms = ½ , -½
½
2 states
2 states
2 states
t t
There are a total of 8 states with n=2
Physics 102: Lecture 25, Slide 18
ACT: Quantum Numbers
How many unique electron states exist with n=5
and ml = +3?
A) 0
B) 4
C) 8
D) 16
E) 50
l
l
l
l
= 0 : ml = 0
= 1 : ml = -1, 0, +1
= 2 : ml = -2,, -1,, 0,, +1,, +2
Only
y
l = 3 and l = 4
have ml = +3
= 3 : ml = -3, -2, -1, 0, +1, +2, +3
ms = ½ , -½
2 states
l = 4 : ml = -4, -3, -2, -1, 0, +1, +2, +3, +4
ms = ½ , -½
2 states
There are a total of 4 states with n=5, ml = +3
Physics 102: Lecture 25, Slide 19
Pauli Exclusion Principle
In an atom with many electrons only one electron
quantum state ((n,, l,, ml, ms)).
is allowed in each q
This explains the periodic table!
Physics 102: Lecture 25, Slide 20
Electron Configurations
#
electrons Atom
Configuration
1
2
H
He
1s1
1s2
3
Li
1s22s1
4
Be
1s22s2
5
B
etc
1s22s22p1
10
Ne
(n=1 shell filled noble gas)
2s shell filled
1s22s22p6
s shells hold up to 2 electrons
Physics 102: Lecture 25, Slide 21
1s shell filled
2p shell filled
(n=2 shell filled noble gas)
p shells hold up to 6 electrons
The Periodic Table
s (l =0)
n = 1, 2
2, 3, ...
p (l =1)
Also s
d (l =2)
f (l =3)
What determines the sequence? Pauli exclusion & energies
Physics 102: Lecture 25, Slide 22
Summary
• Each electron state labeled by 4 numbers:
n = principal
i i l quantum
t
number
b (1,
(1 2,
2 3,
3 …))
l = angular momentum (0, 1, 2, … n-1)
ml = componentt off l (-l
( < ml < l))
ms = spin (-½ , +½)
• Pauli Exclusion Principle explains periodic table
• Shells fill in order of lowest energy.
gy
Physics 102: Lecture 25, Slide 23